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Ashwani Goyal’s Tutorial
TOPICS COVERED
SOLUTION
If two sides of a triangle are the roots of the
(
)
4 x2 − 2 6 x +1 = 0
equation
TRIANGLES
7.
equation 81sin x + 81cos x = 30, then the triangle
can not be
(1) eqilateral
(2) isosceles
(4) right angled
(3) obtuse angled
2
and the
included angle is 60°, then the third side is
(1)
(2)
3
(3) 1/ 3
3/2
(4) 2 3
8.
In a triangle, if the sum of two sides is x and
their product is y such that
( x + z )( x − z ) = y
where z is the third side of
the triangle then the triangle is
(1) equilateral
(2) right angled
(3) obtuse angled
(4) none of these
3.
9.
tan A tan B tan C
=
=
a
b
c
10.
Go
ya
(2)
(3) 1/ 2
l's
(1) a tan A = b tan B = c tan C
(3) tan A + tan + tan C = 3 3
(4) tan A tan B tan C = abc
4.
5.
In a triangle ABC, cos A cos B + sin A sin B sin
C = 1, then the triangle is
(1) equilateral
(2) right angled isoscels
(3) obtuse angled isosceles
(4) none of these
In a triangle ABC, if D is the middle point of BC
and AD is perpendicular to AC, then cos B =
(1) 2 b/a
(2) – b/c
(
(3) b + c
6.
2
2
) / ca
(4)
(c
2
B −C
2
(3) cos ( B + A / 2 )
(2) cos
11.
12.
(4) cos (C + A / 2 )
If n, n + 1, n + 2; where n is any natural
number, represent the sides of a triangle ABC in
which the largest angle is twice the smallest,
then n =
(1) 1
(2) 2
(3) 3
(4) 4
If the median of the triangle ABC through A is
perpendicular to AB, then tan A + 2tan B =
(1) tan C
(2) sin C
(3) cos C
(4) none of these
If
in
a
triangle
ABC,
(1) 26 − sec3 C
+ a ) / ca
B+C
2
( 13 − 1) / 4
(4) (1 − 13 ) / 4
(2)
cos A cos B cos C
=
=
= k then
7
19
25
2
In the besector of angle A of the triangle ABC
makes an angle θ with BC, then sin θ =
(1) cos
In a triangle ABC if sin A sin B = ab / c 2 , then
the triangle is
(1) equilateral
(2) right angled
(3) obtuse angled
(4) none of these
If the sides a, b, c of a triangle are in G.P. and
largest angle exceeds the smallest by 60°, then
cos B =
(1) 1
cos A cos B cos C
=
=
,
In a triangle ABC if
a
b
c
then the which of the following is not true
2
M
at
2.
If the angles of a triangle ABC satisfy the
h
1.
OF
13.
(2) 32 − sec 2 B
(3) 44 − sec3 A
(4) 0
In a triangle the sides are 3, 4, 5 units, then the
−1/ k
25
19
25
−1/ k
7
19
7
−1/ k
is equal to circumradius is
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Ashwani Goyal’s Tutorial
Mathematics - Material
15.
21.
∠OBC = A / 2, ∠OCA = B / 2, ∠OAB = C / 2,
then
sin ( A − (C / 2 )) sin ( B − ( A / 2 )) sin (C − ( B / 2 ))
sin ( A / 2 ) sin ( B / 2 ) sin (C / 2 )
(1) cos ( A / 2 ) cos ( B / 2 ) cos (C / 2 )
(2) sin A sin B sin C
(3) 1
(4) cos A cos B cos C
(2) cos 2 A + cos 2 B + cos 2C = −3 / 2
22.
(3) sin A + sin B + sin C = 2 2
M
at
(1) 2/3
(3) 1/5
23.
l's
Go
ya
(2) 9/4
24.
20.
(1) 3 cos C
(2) 3 sin C
(4) 3 - 3 cos C
(3) 3 cos (A-B)
In a triangle ABC, if A =18°, b - a = 2, ab = 4,
then the triangle is
(1) acute angled
(2) right angled
(3) obtuse angled
(4) isosceles
If in a triangle ABC, C = 60°, then
25.
26.
a+b+c
ab
3
(3)
a+b+c
(2)
a (a + c − b )
b (b + c − a ) is equal to
1 − cos A
1 − cos B
(2)
1 + cos A
1 + cos B
cos 2 ( A / 2 )
(3)
sin 2 ( B / 2 )
(4)
sin 2 A
sin 2 B
In
a
triangle
ABC
if
(1) 143/432
(2) 13/33
(3) 11/39
(4) 12/37
The perimeter of a triangle right angled at C is
70, and the in-radius is 6, than | a – b | =
(1) 1
(2) 2
(3) 8
(4) 9
If for a triangle ABC, a,b and A are given , then
which of the following gives us two such
triangles
(1) a < b sin A
1
1
+
is equal to
a+c b+c
(1)
(2) 3/2
(4) 5
s −a s −b s −c
=
=
, then tan 2 ( A / 2 ) =
11
12
13
(4) cos 2 A + cos 2 B + cos 2 C
In a triangle ABC, if a+b =3c, then
cos A + cos B =
19.
If a triangle ABC,
(1)
sin A sin B + sin B sin C + sin C sin A =
18.
then
tan ( A / 2 ) tan ( B / 2 ) is equal to
(4) cos A + cos B + cos C = 1
In
a
triangle
ABC,
if
cos A + 2 cos B + cos C = 2, then a, b, c are in
(1) A.P.
(2) G.P.
(3) H.P.
(4) none of these
(1) 0
(3) 1
If in a triangle ABC, 5cos C + 6cos B = 4 and
6cos A + 4cos C = 5
cos ( A / 2 ) cos ( B / 2 )
a b c
b c a = 0,
then
17. In a triangle ABC, if
c a b
,
is equal to
4cos A cos B sin C
16.
If O is point inside the triangle ABC such that
h
14.
(1) 2
(2) 2.5
(3) 3
(4) 3.5
In a triangle ABC, the sides are 13, 14, 15 units,
then the in-radius r is
(1) 3
(2) 4
(3) 5
(4) 7
If ABC is not a right angled triangle, then which
of the following is possible
(1) sin 2 A + sin 2 B + sin 2C =
(2) a = b sin A
(3) a > b sin A and a < b
a −b+c
ab
27.
In a triangle ABC, if B = 30° and
c = 3 b, then A can be equal to
(1) 45°
(3) 90°
3
(4)
a+b−c
:2:
Copyright © 2013 GoyalsMath.com .All rights reserved.
(2) 60°
(4) 120°
Ashwani Goyal’s Tutorial
Mathematics - Material
28.
If in a triangle ABC sines of angles A and B
1
1
1
+
+
=
s −a s −b s −c
satisfy the equation 4 x 2 − 2 6 + 1 = 0, then
cos ( A − B ) is equal to
29.
(1) 0
(2) 1/2
(3) 1/ 2
(4)
35.
3/2
If a triangle ABC, A= 2B, then a 2 − b 2 is equal
to
(1) ab
(2) bc
(3) ca
(4) none of these
36.
a 2 + b2
30. In a triangle ABC, if 2
sin (A – B) = 1
a − b2
cos ( A − B ) is equal to
(2) 4 R sin ( B + C ) / 2 
(2) cos 
C π 
− 
 2 4
C π 
− 
 2 4
(4) cot 
C π 
− 
 2 4
(3) 4 R sin ( B / 2 ) sin (C / 2 )
M
at
C π 
− 
 2 4
h
(1) 4 R sin ( A / 2 )
(3) tan 
37.
2
l's
In a triangle ABC, if the median AD makes an
angle θ with AC and AB = 2AD then sin θ =
(1) sin A
(2) sin B
(3) sin C
(4) none of these
In
a
triangle
ABC,
if
Rr (sin A + sin B + sin C ) = 96, then area of
33.
34.
the triangle i units is equal to
(1) 24
(2) 48
(3) 96
(4) 192
A quadrilateral ABCD in which AB = a, BC = b
CD = c and DA = d is such that one circle can
be insribed in it and another circle
circumscribed about it, then cos A =
ad − bc
(1)
ad + bc
ab − cd
(2)
ab + cd
ad + bc
(3)
ad − bc
ab + cd
(4)
ab − cd
In a triangle ABC if cot
(4) none of these
If the area of the triangle ABC is
a 2 − (b − c ) , then its circumradius R =
Go
ya
32.
the equation x 2 + px − 1, then the triangle is
obtuse angled
(1) if p >1
(2) if p < 1
(3) for finite values of p
(4) for all values of p
The distance of the incentre of the triangle
ABC from A is
and the triangle is not right angled, then
(1) sin 
31.
(1) – 1
(2) 0
(4) 2
(3) 1
In tangents of two angles of a triangle satisfy
38.
39.
A
B
B
cot
= c, cot cot
2
2
2
40.
C
C
A
= a and cot cot = b, then
2
2
2
(1) ( a / 6 ) sin
(2) ( a /16 ) cos ec
2
( A / 2)
(3) (b /16 ) sin
2
(B / 2)
(4) ( c /16 ) sin
2
(C / 2 )
In a triangle ABC,
cos B + cos C
=
1 − cos A
(1)
b+c
1− a
(2)
bc
1− a
(3)
b+c
a
(4)
a
b+c
In a triangle ABC,
r1 + r2
=
1 + cos C
( a + b ) / c∆
(1) 2ab / c∆
(2)
(3) abc / 2∆
(4) abc / ∆ 2
(
)
2
2
2
In a triangle ABC, a − b − c tan A +
(a
:3:
( A / 2)
2
2
− b 2 + c 2 ) tan B is equal to
Copyright © 2013 GoyalsMath.com .All rights reserved.
Ashwani Goyal’s Tutorial
Mathematics - Material
)
(
)
2
2
2
(2) a + b + c tan C
(
(3) b + c − a
2
2
2
47.
) tan C
(
3
2
In a triangle ABC, if cot A = x + x + x
cot B = ( x + x −1 + 1) and
1/ 2
cot B = ( x −3 + x −2 + x −1 )
−1/ 2
42.
(1) sin ( 2C − A) = sin ( B / 4 )
cot
)
1/ 2
C
(2) sin ( A − C ) = sin ( B / 2 )
(3) sin ( A + C ) = cos 2 B
=
(4) cos ( A − C ) = sin ( B / 2 )
then the triangle
is
(1) isosceles
(2) obtuseangled
(3) right angled
(4) none of these
In a triangle ABC, if a is the arithmetic mean
48.
(
sin 3 B + sin 3 C
is equal to
sin A sin B sin C
45.
49.
l's
1 − sin A
AD
(4)
cos B
50.
In a triangle ABC r1r2 + rr3 =
ab
(1)
c
(2) abc
(3) ab
(4)
3 +1
(2) 2 + 2 3
(3) 2 2
(4) 4
If A is the area and 2 s,the sum of three sides
of a triangle, then
(1) A ≤
s2
3 3
(2) A ≤
(3) A >
s2
3
(4) none of these
s2
2
In a triangle ABC, if a = 5, b = 4 and c = 3, D
and E are the points on BC such that BD = DE
= EC. If ∠DAE = θ , then tan θ =
(1) 3/8
(2) 2/3
(3) 18/25
(a + b ) / c
51.
(4) 1/ 3
In a triangle ABC,the sides a, b, c are
If d1 , d 2 , d 3 are the diameters of the three
respectively 13,14,15.If r1 is the radius of the
escribed circles of a triangle ABC, then
escribed circle touching BC and the sides AB
d1 d 2 + d 2 d3 + d3 d1 =
and AC produced, then r1 is equal to
(1) 10.5
(3) 14
a b c
+ +
b c a
(1) ab + bc + ca
(2)
(3) ( a + b + c )
(4) none of these
2
46.
(1)
Go
ya
44.
is
(1) 0
(2) 1
(3) 2
(4) 4
If D is a point of the base BC of an isoceles
triangle ABC such that AD is perpendicular to
AC, then a =
(1) 2 b
(2) 2b cos B
1 + cos A
AD
(3)
sin B
)
M
at
between any two positive real numbers then
The angles of a triangle ABC satisfy the
relations 3B – C = 30° and A+2B = 120°.If the
perimeter
of
the
triangle
is
2 3 + 3 + 2 , the largest side of the triangle
and b, c (b ≠ c ) are the two geometric means
43.
3 : 2, then
=
(4) none of these
41.
mid-point of BC, then cos ∠ADC is equal to
(1) 7/25
(2) –7/25
(3) 24/25
(4) – 24/25
In a triangle ABC, if A,B,C are in A.P. and b : c
h
(
2
2
2
(1) a + b − c tan C
52.
If in a triangle ABC, r1 = 2r2 = 3r3 ; D is the
:4:
(2) 12
(4) 4
The distance of the orthocentre of V ABC from
its vertex A is
(1) 2R sin A
(2) 2R sin B cos B
(3) 2R cos A
(4) 2R sin B sin C
Copyright © 2013 GoyalsMath.com .All rights reserved.
Ashwani Goyal’s Tutorial
Mathematics - Material
53.
If R, r denote respectively the circum-radius
and in-radius of a triangle, then the distance
between the circumcentre and the incentre of
the triangle is
(2) R 2 − 2 Rr
(1) 2Rr
54.
(3) R 2 + 2 Rr
(4) R + r
If R is the circum radius of the triangle ABC
and r1 is the radius of the escribed circle which
touches BC and the two sides AB and AC
produced, then the distance of the circumcentre
from the centre of the escribed circle is
(2) R 2 − 2 Rr1
(3) R 2 + 2 Rr1
(4) R + r1
M
at
(1) R 2 − 8 R 2 sin A cos B cos C
h
If R is the circum-radius of a triangle ABC then
the distance of the orthocentre of the triangle
from its circumcentre is
(2) R 2 − 8R 2 cos A sin B cos C
(3) R 2 − 8 R 2 sin A sin B sin C
l's
(4) R 2 − 8 R 2 cos A cos B cos C
Go
ya
55.
(1) 2Rr1
:5:
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Ashwani Goyal’s Tutorial
Mathematics - Material
EXERCISES
(1) A + B - C =90°
(2) The triangle is acute angled
(3) A,B,C are in A.P.
(4) the triangle is obtuse angled
2.
In a triangle ABC, if tan
8.
B −C 3
A
= cot ,
2
5
2
(
(2) ( c 2 / 4 ) sin B sin 2 B
2
(1) 2b 2 sin A
(2) 2c 2 sin A
(3) (1/ 8 ) c sin A
(4) 2c 2 cos A
2
10.
l's
Go
ya
11.
(1) sin B
(2) cos B
(3) sin [(A + C)/2]
(4) sin[(A - C)/2]
a right angled triangle of maximum area is
inscribed in a circle of radius R, the radius of the
incircle is
2R
(2)
(
12.
)
2 −1 R
(3) R/2
(4) none of these
If x, y, z are perpendiculars from the angular
points of a V ABC upon the opposite sides a, b,
c respectively, then
bx cy az
+ +
is equal to
c
a b
(1) b cos B + c cos C + a cos A
(
2
2
2
(2) 2 R sin A + sin B + sin C
(
13.
)
2
2
2
(3) 2 R cos A + cos B + cos C
(
)
2
(4) b / 4 sin 2C
bc sin 2 A
=
In a triangle ABC
cos A + cos B cos C
(1) b 2 + c 2
In a triangle ABC, if s - a, s - b, s - c are in G.P.,
(1)
6.
9.
(4) 8cos α (1 − cos α )
sin 2 A + sin 2 C
=
then
sin A + sin C
)
M
at
8cos α
(2)
1 − cos α
8cos α
(3)
1 + cos α
5.
(
2
(3) c / 4 sin 2 B
In an isosceles triangle with base angle α and
lateral side 4, Rr =
(1) 8cos α
4.
)
2
2
(1) c / 4 cos B sin 2 B
then D =
3.
If a chord of length 1 unit subtends an angle θ
at the circumference of a circle whose radius is
R then R sin θ =
(1) 2
(2) 1/2
(3) 1/3
(4) 1/4
In a triangle ABC, AD is perpendicular to BC
and DE is perpendicular to AB, then area of
∆ADE =
h
1.
7.
a
b
c
= then
In a triangle ABC if =
1
3 2
(2) bc
(3) a 2
(4) a 2 + bc
If the sides of a triangle are 17, 10, 21, then the
radius of the circum-circle is approximately
(1) 10.5
(2) 10.6
(3) 10.7
(4) 10.8
In a triangle PQR, sin P, sin Q, sin R are in A.P.,
then
(1) the altitudes are in A.P.
(2) the altitudes are in H.P.
(3) the medians are in A.P.
(4) the medians are in H.P.
In a triangle ABC, sin A, sin B, sin C are
irrational numbers, then
(1) sides of the triangle must be irrational numbers
(2) medians of the triangle must be irrational
numbers
(3) altitudes of the triangle must be irratinal numbers
(4) none of these
In a triangle ABC, Let ∠C = π / 2, if r is the
inradius and R is the circum radius of the
triangle, then 2 ( r + R ) is equal to
)
(4) none of these
:6:
(1) a + b
(2) b + c
(3) c + a
(4) a + b + c
Copyright © 2013 GoyalsMath.com .All rights reserved.
Ashwani Goyal’s Tutorial
Mathematics - Material
14.
In a triangle ABC, 2ac sin [ (A - B + C)/2] is
equal to
(1) a 2 + b 2 − c 2
15.
20.
(2) c 2 + a 2 − b 2
(3) b 2 − c 2 − a 2
(4) c 2 − a 2 − b 2
In
a
triangle
ABC,
cos A cos B cos C
=
=
and
a
b
c
(1)
(3) r − 2 R
a = 2 3 cm,
21.
(3) 4 cm 2
(4) 3 3 cm 2
If cos A =
sin B
,
2sin C then ∆ ABC is
(4)
1 1
−
r 2R
In a griangle ABC
b2 − c2
c2 − a2
a2 − b2
cos A +
cos B +
cos C i s
a
b
c
22.
equal to
(1) 0
(2) 1
2
2
2
(4) a b c
(3) a + b + c
If the angles, A, B, C of triangle ABC are in
arithmetical progression then
h
(1) equilateral
(2) isosceles
(3) right angled
(4) none of these
In an equilateral triangle the in-radius,
circumradius and one of the ex-radii aer in the
ratio
(1) 2 : 3: 5
(2) 1 : 2 : 3
(3) 1 : 3 : 5
(4) 3 : 5 : 7
(1) tan A + tan C − 3 tan A tan C = 3
(2) tan A + tan C + 3 tan A tan C = 3
(3) tan A + tan C − 3 tan A tan C = − 3
(4) tan A + tan C + 3 tan A tan C = − 3
If the angles of a ∆ are 30° and 45° and the
included side is
(
)
3 + 1 cm, the area of the
triangle is
)
3 + 1) cm
3 + 1) cm
3 + 1) cm
3 + 1 cm
Go
ya
(
(2) (1/ 4 ) (
(3) (1/ 6 ) (
(4) (1/ 8 ) (
(1) (1/ 2 )
19.
(2) 2R − r
M
at
18.
(2) 3cm 2
l's
17.
(1) 2 3 cm 2
1 1
−
2R r
r1 r2 r3
+ +
is equal to
bc ca ab
if
then the area of the triangle is
16.
In any triangle ABC ,
In a griangle ABC if
24.
(B+C) is equal to
(1) cos B cos C
(2) coa A cos C
(3) cos A cos B
(4) sin B sin C
If the angles of a triangle are in the ratio 1: 3 : 5
and θ denotes the smallest angle, then the ratio
of the largest side to the smallest side of the
triangle is
2
2
2
(1)
3 sin θ + cos θ
(2)
2sin θ
3 cos θ − sin θ
2sin θ
(3)
cos θ + 3 sin θ
(4)
2sin θ
3 cos θ + sin θ
2sin θ
2
If α , β , γ are the lengths of the altitudes of a
triangle ABC, then
(1) π / ∆
(3) π
1
1
1
+ 2 + 2 is equal to
2
α
β
γ
(2) ∆ / π
(4) none of these
25.
where π = cot A + cos B + cot C and ∆ is the
of the triangle
In a triangle ABC, (b + c ) / a is equal to
(1)
(3)
:7:
cos A tan C
=
, then sin
a
c
23.
sin ( B − C ) / 2
sin ( B + C ) / 2 
sin ( B + C ) / 2 
sin ( B − C ) / 2
Copyright © 2013 GoyalsMath.com .All rights reserved.
(2)
(4)
cos ( B − C ) / 2 
cos ( B + C ) / 2 
cos ( B + C ) / 2 
cos ( B − C ) / 2 
Ashwani Goyal’s Tutorial
Mathematics - Material
26.
27.
28.
In a triangle ABC, 1– tan (A/2) tan (B/2) is
equal to
(1)
2a
b+c−a
(2)
2b
c+ a −b
(3)
2c
a+b−c
(4)
2c
a+b+c
33.
34.
If in a triangle ABC, b + c = 3a, then cot (B/2)
cot (C/2) is equal to
(1) 1/2
(2) 1
(3) 2
(4) non of these
The angles of a triangle ABC are in A.P. The
largest is twice the smallest and the median to
the largest side divides the angle at the vertex in
the ratio 2 : 3. If length of the median is
(1) a cos[( B − C ) / 2] (2) a sin[( B − C ) / 2]
(3) a cos[( B + C ) / 2] (4) a sin[( B + C ) / 2]
35.
(4)
The expression
a2 + b2 − c2 − d 2
b2 + c2 − d 2 − a 2
(1)
2 ( ab + cd ) (2)
2 (bc + da )
h
(3) 8 sin 32°
3 sin 40°
( a + b + c )(b + c − a )
(c + a − b )( a + b − c ) is equal
to
2
(3) cot ( A / 2 )
2
(4) tan ( A / 2 )
l's
2
(2) sin ( A / 2 )
37.
If the median AD of a triangle ABC divides the
angle ∠BAC in the ratio 1 : 2, then sinB/sinC is
equal to
Go
ya
30.
c2 + d 2 − a2 − b2
d 2 + a 2 − b2 − c 2
(3)
2 ( cd + ab ) (4)
2 ( da + bc )
36.
2
(1) cos ( A / 2 )
(1) 2 cos ( A / 3)
In
2 ( p + q )(1 − pq )
(2) (1/ 2 ) sec ( A / 3)
a
b sin
32.
2
triangle
(1 + p )(1 + q )
2
(1) sin A
(3) sin C
ABC,
( A / 2 ) + a sin ( B / 2 ) is equal to
38.
2
(1) s – c
(2) s + c
(3) s − ( a + b )
(4) none of these
2 ab
C
cos
a −b
2
(2) sin B
(4) sin A + sin B
In a triangle ABC, a 2 cos 2 B + b 2 cos 2 A is
equal to
(3) a 2 + b 2 + 4ab cos A sin B
(4) a 2 + b 2 − 4ab cos A sin B
2 ab
C
tan
(2)
a −b
2
(4)
is equal to
(2) a 2 + b 2 − 4ab sin A sin B
tan θ is equal to
(3)
2
(1) a 2 + b 2 + 4ab sin A sin B
In a triangle ABC , if c = ( a − b ) secθ , then
2 ab
C
sin
(1)
a−b
2
In a triangle ABC, sin A + sin B + sin C is
maximum when the triangle is
(1) right angled
(2) isosceles
(3) equilateral
(4) obtuse angle
In
a
triangle
ABC,
if
tan ( A / 2 ) = p, tan ( B / 2 ) = q, then
(3) (1/ 2 ) sin ( A / 3) (4) 2 cos ec ( A / 3)
31.
In a cyclic quadrilateral ABCD; a, b, c, d denote
the length of the sides AB, BC, CD and DA
respectively, then cos A is equal to
M
at
29.
(2) 2 sin 48°
In a triangle ABC, (b − c ) cos ( A / 2 ) is equal
to
2 3 cm , length of the largest side is
(1) 2 sin 32°
The vertical angle of a triangle is divided into
two parts, such that the tangent of one part is 3
times the tangent of the other and the
difference of these parts is 30°, then the triangle
is
(1) isosceles
(2) right angled
(3) obtuse angled
(4) none of these
39.
If the angles A and B of the triangle ABC
satisfy
the
relation
sin A + sin B = 3 (cos B − cos A ) then they
2 ab
C
sec
a−b
2
:8:
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Ashwani Goyal’s Tutorial
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41.
42.
43.
(1) 50 2 m 2
(2) 100 2 m 2
(3) 200 2 m 2
(4) 100 m 2
49.
(2) k 2 − 6 k + 1 = 0
47.
(4) 7 3 / 22
If A, B, C, D are the angles of a quadrilateral,
(4)
50.
51.
Go
ya
52.
(2) 7/48
(4) 7/12
53.
tan A + tan B + tan C + tan D
is equal to
cot A + cot B + cot C + cot D
∑ tan A tan B tan C
If in a triangle ABC, the line joining the
circumcentre 0 and the incentre I is parallel to
BC, then
(1) r = R cos A
(2) r = R sin A
(3) R = r cos A
(4) R = r sin A
If the sides of a triangle are 17, 10 and 21, then
the largest ex-radii of the triangle is
(1) 6
(2) 12
(3) 28
(4) none of these
If the area of a triangle is 96, and the radii of the
escribed circles are 8, 12, 24, then the largest
side of the triangle is
(1) 16
(2) 18
(3) 20
(4) 32
If R is the circumradius of a triangle ABC then
the area of its pedal triangle is
(1) (1/ 2 ) R sin A sin B sin C
cos A cos B cos C a
+
+
= , then A is
a
b
c
bc
46.
(3) 2 3 /11
(3) tan 2 A + tan 2 B + tan 2 C + tan 2 D
2
then tan (θ / 2 ) is equal to
45.
(2) 28 3 /11
(2) cot A cot B cot C cot D
(4) 3 − 2 2 < k < 3 + 2 2
The sides of a triangle are in the ratio 5 : 8 : 11 .
If θ denotes the largest angle of the triangle
(1) 1/21
(3) 7/3
In a triangle ABC if
(1) 28/11
(1) tan A tan B tan C tan D
l's
(3) k 2 − 6 k + 1 ≤ 0
221
In a triangle ABC, if b = 7, c = 4 and A = 120°
then the length of the bisector of angle A is
then
In a triangle ABC if A = π / 4 and tan B tan C
= k, then k must satisfy
(1) k 2 − 6 k + 1 ≥ 0
44.
48.
(3) π / 4
(4) π / 2
In a triangle ABC, right angled at C, the sum of
the tangents of the other two angles is 169/60.
The tangent of the larger of these two is equal
to
(1) 5/12
(2) 12/5
(3) 12/13
(4) 13/5
The sides of a triangle are 17, 25, 28, The length
of the largest altitude is
(1) 15
(2) 84/5
(3) 420/17
(4) 210/17
Two angles of a triangular field are 22.5° and
45°, and the length of the sides opposite to the
later is 200 metres. The area of the field is
(4)
M
at
40.
(3) 14
(2) π / 3
h
differ by
(1) π / 6
2
2
(2) (1/ 2 ) R sin 2 A sin 2 B sin 2C
(1) an acute angle
(2) an obtuse angle
(3) a right angle
(4) equal to B - C
If one base angle of a triangle is five times the
other and half the angle at the vertex, then base
of the triangle is equal to
(1) the height
(2) half the height
(3) twice the height (4) five times the height
In a triangle ABC, the median AD is
(3) (1/ 2 ) R cos A cos B cos C
2
2
(4) (1/ 2 ) R cos 2 A cos 2 B cos 2C
54.
perpendicular to AC; if b = 5, c = 11, then a is
equal to
(1) 10
(2) 12
If R is the circumradius of the triangle
ABC,then the circumradius of its pedal triangle
is
(1)
R
(3) R/3
:9:
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(2) R/2
(4) none of these
Ashwani Goyal’s Tutorial
Mathematics - Material
If S is the circumcentre of the triangle whose
58.
the incentre and excentres of the triangle, then
If R is the circumradius of the triangle ABC
then the circumradius of the ex-central triangle
is equal to
(1) R/2
(2) 2R
SI 2 + SI 21 + SI 2 2 + SI 23 is equal to
(3) R 2
circumradius is R; I , I1 , I 2, I 3 are respectively
(2) 8R 2
(1) 4R 2
56.
59.
(3) 12R 2
(4) none of these
If I is the incentre of the triangle ABC, then
aAI 2 + bBI 2 + cCI 2
(1) abc
(2) a 2 + b 2 + c 2
60.
(3) (1/ 2 )( a + b + c ) (4) a 2b 2 c 2
l's
M
at
Let ABC be a triangle I1, I2, I3 its ex-centres :
then orthocentre of the Ex-central triangle I1
I2 I3 is the
(1) centroid
(2) circumcentre
(3) in-centre
(4) orhtocentre
: 10 :
R
If I is the incentre of a triangle whose in radius
and circumradius are r and R respectively; I1 I2
I3 is its ex-central triangle, then II1 . II2 . II3 is
equal to
(1) R 2 r
(2) 16 R2
(3) Rr 2
(4) 16Rr2
If the radius of the circumcircle of a triangle is
twice the radius of the in-circle of the triangle
then the triangle is
(1) right-angled
(2) isosceles
(3) equilateral
(4) obtuse angled
Go
ya
57.
(4)
h
55.
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Ashwani Goyal’s Tutorial
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TOPICS COVERED
HEIGHTS
3.
Two flagstaffs stand on a horizontal plane. A
and B are two points on the line joining their feet
between them. The angles of elevation of the
tops the flagstaffs as seen from A are 30° and
60° and as seen from B are 60° and 45°. If AB
is 30 m, the distance between the flagstaffs in
metres is
(1) 30 + 15 3
(2) 45 + 15 3
(1) sin θ = 5 /12
(2) cos θ = 5 /12
(3) 60 − 15 3
(4) 60 + 15 3
(3) sin θ = 3 / 4
(4) cos θ = 3 / 8
(3) 300 2 − 3 m
(4) 400 m
7.
h
(2) 300 2 + 3 m
M
at
(1) 200 m
In a cubica hall A B C D P Q R S with each
side 10 m G is the centre of the wall B C R Q
and T is the mid point of the side AB. The angle
of elevation of G at the point T is
−1
(3) tan
5.
A pole stands vertically, inside a triangular park
ABC. If the angle of elevation of the top of the
pole from each corner of the park is same, then
in ∆ ABC , the foot of the pole is at the
(2) circumcentre
(1) centroid
(3) incentre
(4) orthocentre
A man observes that the angle of elevation of
the top of a tower from a point P on the ground
is θ . He moves a certain distance towards the
food of the tower and finds that the angle
elevation of the top has doubled. He further
moves a distance 3/4 of the previous and finds
that the angle of elevation is three times that at
P. The angle θ is given by
(1) sin
4.
6.
8.
l's
2.
A flagstaff stand in the centre of a rectangular
field whose diagonal is 1200 m, and subtends
angles 15° and 45° at the mid points of the sides
of the field. The height of the flagstaff is
−1
Go
ya
1.
DISTANCES
AND
(1/ 3 )
(1/ 3 )
(2) cos
(4) cot
−1
−1
(1/ 3 )
(1/ 3 )
(1) cos
Two vertical poles 20 m and 80 m high stand
apart on a horizontal plane. The height of the
point of intersection of the ines joining the top of
each pole to the foot of the other is
(1) 15 m
(2) 16 m
(3) 18 m
(4) 50 m
A man from the top of a 100 metres high tower
sees a car moving towards the tower at an
angle of depression of 30°. After some time, the
angle of depression becomes 60°. The distance
(in metres) travelled by the car during this time
is
(1) 100 3
(2) 200 3 / 3
(3) 100 3 / 3
(4) 200 3
A and B are two points 30 m apart in a line on
the horizontal plane through the foot of a tower
lying on opposite sides of the tower. If the
distance of the of the tower from A and B are
20 m and 15 m respectively, the angle of
elevation of the top of the tower at a is
(3) cos
9.
(3) tan
: 11 :
( 43 / 48)
(2) sin
−1
( 29 / 36 )
(4) sin
−1
(43 / 48)
−1
(29 / 36 )
A vertical pole subtends P on the ground. The
angle subtended by the upper half of the pole at
the P is
(1) tan
10.
−1
−1
(1/ 4)
(2) tan
−1
(1/ 8)
(4) tan
−1
(2 / 9)
−1
( 2 / 3)
An aeroplane flying at a height of 3000 m above
the ground passes vertically above another at
an instant when the angles of elevation of the
two planes from the same point on the ground
are 60° and 45° respectively. The height of the
lower plane the ground is
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(3) 500 m
(4) 1500
(
(3) (
)
3 / 2 ) h tan α
3 / 2 h cot α
(
17.
(2 / 3 ) h cot α
(2 / 3 ) h tan α
(1) 3a 2 = 2h 2
(2) 2a 2 = 3h 2
(3) a 2 = 3h 2
(4) 3a 2 = h 2
19.
Go
ya
)
3 − 1)
(
h=a(
(2) h = a
(4)
)
3 − 1)
3 +1
The angle of elevation of a vertical tower
standing inside a triangular field at the vertices
of the are each equal to θ . If the length of the
sides of the field are 30 m, 50 and 70 m, the
height of the tower is
(1) 70 3 tan θ m
(
18.
(4) 25 3 m
3 +1
(2)
20.
(70 / 3 ) tan θ m
)
(3) 50 / 3 tan θ m (4) 75 3 θ m
16.
(4)
(a + b )
2
A monument ABCD stands an a level grond.
At point P on the ground the portions AB, AC,
α + β + γ = 1800 then x2 is equal to
The angle of elevation of the top C of a vertical
tower CD of height h from a point a in the
horizontal plane is 45° and from a point B at a
distance a from A on the line making an angle
30° with AD, it is 60°, then
(
(3) a = h (
)
(2) a 2 + b 2
AD subtend angles α , β , γ respectively. If
AB = a, AC = b, AD=c, AP = x and
A pole 50 m high stands a building 250 m high.
To an observer at a height of 300 m, the building
and the pole subtend equal angles. The
horizontal distance of the observer from the
pole is
(1) 25 m
(2) 50 m
(1) a = h
15.
(4)
1 2
a + b2 )
(
2
2
2
(3) 2 a + b
If each side of length a of an equilateral triangle
subtends an angle of 60° at the top of tower h
situated at the centre of the triangle, then
(3) 25 6 m
14.
(2)
(1)
(1)
a
a+b+c
(2)
b
a+b+c
(3)
c
a+b+c
(4)
abc
a+b+c
M
at
13.
)
3 +1 m
A pole of height h stands at one corner of a
park in the shape of an equilateral triangle. If α
is the the angle which the pole subtends at the
midpoint of the opposite side, the length of each
side of the park is
(1)
12.
(
same angle 45° at a point on the line joining their
feet, the square of the distance between their
tops is
h
(2) 1000 / 3 m
l's
11.
(1) 1000 3 m
Two vertical poles of height a and b subtend the
21.
: 12 :
In a triangular plot ABC with BC = 7cm,
CA=8m and AB = 9m. A lamp post is situated
at the middle point E of the side AC and
subtends an angle tan–1 3 at the point B, the
height of the lamp post is
(1) 21 m
(2) 24 m
(3) 27 m
(4) can not be determined
A vertical tower CP subtends the same angle
θ , at point B on the horizontal plane through C,
the of the tower, and at point A in the vertical
plane. If the triangle ABC is equilateral with
length of each side equal to 4m, the height of the
tower is
(1) 8 3 m
(2) 4 3 / 3 m
(3) 4 3 m
(4) 8 / 3 m
Two objects at the points P and Q subtend an
angle of 30° at a point A. Lengths AR = 20 m
and AS = 10 m are measured from A at right
angles to AP and AQ respectively. If PQ
subtends equal angles of 30°, at R and S, then
length of PQ is
(1)
300 − 200 3
(2)
500 − 200 3
(3)
500 3 − 200
(4)
300
From a ship at sea it is observed that the angle
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Ashwani Goyal’s Tutorial
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25.
26.
(1)
18 × 93
(2)
36 × 93
(3)
34 × 93
(4)
34 × 36
28.
(
)
2 +1 m
(4) 5
(4) cos −1 2h / x
(
h
The elevation of a stteple at a place due south
of it is 45° and at a place B due west of A the
elevation is 15°. If AB = 2a, the height of the
steeple is
(1) a
)
3 −1
(1) 500 6 m
(2) 500 3 m
(3) 250 6 m
(4) 250 3 m
31.
A lamp post standing at a point a on a circular
path of radius r subtends an angle α at some
point B on the path, and AB subtends an angle
of 45° at any other point on the path, then height
of the loampost is
(3)
2 r cot α
(2)
2r tan α
(4)
(r / 2 ) tan α
(1/ 2 )cot α
(2) a
2
 14 − 14 
a
(3) 3 − 3 


30.
)
3+2 m
x / ( 2 − h ) (2) tan −1 3 − 2h / x
−1
(3) sin −1 2h / x
29.
(
From the top of a cliff x m high, the angle of
depression of the foot of a tower is found to be
double the angle of elevation of the tower. If the
height of the tower is h, the angle of elevation is
(1) sin
From a point on the horizontal plane, the
elevation of the top of a hill is 45°. After
walking 500 m towards its summit up a slope
inclined at an angle of 15° to the horizontal the
elevatio is 75°, the height of the hill is
(1)
27.
(3) 10
M
at
24.
the ground at a point 10 m from the foot of the
tree and makes an angle 45° with the ground.
The entire length of the tree was
(1) 15 m
(2) 20 m
l's
23.
Go
ya
22.
subtended by feet A and B of two light houses,
at the ship is 30°. the ship sails 4 km towards A
and this angle is then 48°, the distance of B
from the ship at the second observation is
(1) 6.460 km
(2) 6.472 km
(3) 6.476 km
(4) 6.478 km
A man on the ground observes that the angle of
elevation of the top of a tower is 68° 11’, and a
flagstaff 24 m high on the summit of the tower
subtends an angle of 2° 10’ at the observer’s
eye. If tan 70° 21’ = 2.8 and cot 68° 11’ = 0.4
the height of the tower is
(1) 120 m
(2) 168 m
(3) 200 m
(4) 300 m
A statue, standing on the top of a pillar 25 m
high subtends an angle whose tangent is 0.125
at a point 60 m from the foot of the pillar. The
best approximation for the height of the statue is
(1) 9.28 m
(2) 9.29 m
(3) 9.30 m
(4) 10 m
A tower BCD surmounted by a spire DE stands
on a horizontal plane. At the extremity a of
horizontal line Ba it is found that BC and DE
subtend equal angles. If BC = 3 m, CD = 28 m
and DE = 5 m, then BA is equal to
(
)
3 +1
2
 14 − 14 
a
(4) 3 + 3 


A river flows due NOrth, and a tower stands on
its left bank. From a point A upstream and on
the same bank as the tower, the elevation of the
tower is 60°, and from a point B just opposite A
on the other bank the elevation is 45°. If the
tower is 360 m high, the breadth of the river is
(1) 120 6 m
(2) 240 3 m
(3) 240 3 m
(4) 240 6 m
The top of a pole, placed against a wall at an
angle α with the horizon, just touches the
coping, and when its foot is moved a m, away
from the the wall and its angle of inclination is
β , it rests on the still of a window;
(
)
(
)
(
)
(
)
(1) a sin (α + β ) / 2 (2) a cos (α + β ) / 2
(3) a cot (α + β ) / 2 (4) a tan (α + β ) / 2
32.
A tree is broken by wind, its upper part touches
: 13 :
OAB is a triangle in the horizontal plane through
the foot P of the tower at the middle point of the
side OB of the triangle. If OA = 2m, OB = 6m,
AB = 5 m and ∠ AOB is equal to the angle
subtended by the tower at A then the height of
the tower is
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(2)
(3)
11× 25
39 × 2
(4) none of these
36.
11× 39
25 × 2
centre of the balloon from the eye is β , the
height h of the centre of the balloon of the
balloon is given by
(1)
If two vertical towrs PQ and RS of lengths a
and b (a > b) respectively subtend the same
angle α at a point a on the line joining their feet
P and R in the horizontal plane and angles
β and γ at another point B on this line nearer
the towers on the side of the towers as A, then
r sin α
37.
sin γ
(3)
sin α
(4)
b sin α
(b − a ) sin α
b sin γ
(3)
38.
The angle of elevation of a clound at a height h
above the level of water in a lake is α , and the
angle of depression of its image in the lake is
AB 1
= sin α
BQ 2
(4)
AB 1
= cos ec α
BQ 2
Two poles of height a and b stand at the centres
of two circular plots which touch each other
externally at a point and the two poles subtend
angles of 30° and 60° repectively at this point,
then distance between the centres of these
plots is
(3a + b ) /
β . The height of the clound above the surface
(1) a + b
(2)
of the lake is
(3) ( a + 3b ) / 3
(4) a 3 + b
(3)
Go
ya
h (cot α + cot β )
h ( tan α + tan β )
(1)
(2)
cot β − cot α
tan α − tan β
35.
r sin α
(4) sin β / 2
( )
A tower PQ subtends an angle α at a point A
on the same level as the foot Q of the tower. It
also subtends the same angle α at a point B
where AB subtends the angle α with AP then
(1) AB = BQ
(2) BQ = 2AQ
(b − 1) sin γ
l's
34.
(2)
(2) r sin b sin a
M
at
b sin α
r sin β
sin α
(3) sin α / 2
( )
sin ( β − γ )
sin ( β − α ) is equal to
(1) a − b sin γ
(
)
a balloon of radius r subtends an angle α at the
eye of an observer and the elevation of the
h
33.
(1)
11× 39
25 × 3
h sin (α + β )
sin ( β − α )
(4)
39.
h sin (α − β )
sin (α + β )
A person standing on the ground observes the
angle of elevation of the top of a tower to be
30°. On walking a distance a in a certain
direction, he finds the elevation of the top to be
the same as before. He then walks a distance
5a/3 at right angles to his former direction, and
finds that the elevation of the top has doubled.
The height of the tower is
(1) a
(3)
6/5 a
(2)
85 / 48 a
(4)
48 / 85 a
The angle of elevation of the top of a tree at a
point B due south of it is 60° at a point C due
north of its is 30°. D is a point due north of C
where the angle of elevation is 15°, then given
31
40.
: 14 :
3
8
and BC × CD = 22 × 32 × 19 × 11, the
11
height of the tree is
(1) 33
(2) 38
(3) 57
(4) 88
n poles standing at equal distances on a straight
road subtend the same angle α at a point O on
the road. If the height of the largest pole is h
and the distance of the foot of the smallest pole
from O is a, the distance between two
consecutive poles is
(1)
h sin α − a cos α
h cos α − a cos α
(2)
( n − 1) sin α
( n − 1) cosα
(3)
h cos α − a sin α
h sin α − a cos α
(4)
( n − 1) sin α
( n − 1) cos α
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Mathematics - Material
EXERCISES
(1) 50 3
(2) 100 2
(3) 100 3
(4) 100
)
3 −1
The angle of elevation of the top of a tower at
the top and the foot of a pole 10 m high are 30°
and 60° respectively. The height of the tower is
(1) 15 m
(2) 20 m
(3) 10 3 m
3.
(
8.
(4) 25 3 m
A tower subtends an angle α at a point A on
the ground, and the angle of depression of its
foot from a point B just above a and at distance
b from a, is β . The height of the tower is
(2) b tan α cot β
(3) b cot α cot β
(4) b cot α tan β
(
(3) 500 (
6.
10.
A person walking along a straight road
observes that at two points 1 km apart, the
angles of elevation of a pole in front of him are
30° and 75°. The height of the pole is
(1) 250
5.
l's
(1) b tan α tan β
(1) tan
Go
ya
4.
9.
)
2 + 1) m
3 +1 m
(
500 (
(2) 250
(4)
From the top of a tower 100 m height, the
angles of depression of two objects 200 m apart
on the horizontal plane and in a line passing
through the foot of the tower and on the same
side of the tower are 45° – A and 45° + A, then
angle A is equal to
(1) 15°
(2) 22.5°
(3) 30°
(4) 35°
An observer finds that the angular elevation of
a tower is A, on advancing 3m towards the
tower the elevation is 45° and on advancing 2m
nearer, the elevation is 90° – A, the height of the
tower is
(1) 1 m
(2) 5 m
(3) 6 m
(4) 8 m
ABC is a triangular park with all sides equal. If
a pillar at A subtends on angle of 45° at C, the
angle of elevation of the pillar at D, the middle
point of BC is
M
at
2.
The angle of elevataion of the top of an
incomplete vertical pillar at a horizontal distance
of 100 m from its base is 45°. If the angle of
elevation of the top of the pillar after completion
at the same point is 60°, then the height to be
increased for the completion of the pillar in
metres is
h
7.
1.
−1
(
(
(4) ( 25 / 2 ) (
(3) ( 25 / 2 )
11.
: 15 :
(2) tan
−1
(2 / 3 )
(4) tan −1 3
A kite is flying with the string inclined at 75° to
the horizon. If the length of the string is 25 m,
the height of the kite is
(
(2) ( 25 / 4 ) (
)
2 − 1) m
)
(3) cot −1 3
(1) ( 25 / 2 )
3 −1 m
If a flagstaff subtends the same angle at the
points A, B, C and d on the horizontal plane
through its foot, then ABCD is
(1) square
(2) cyclic quadrilateral
(3) rectangle
(4) none of these
From a point on the ground 100 m away from
the base of building, the angle of elevation of the
top of the building is 60°. Which of the following
is the best approximation for the height of the
building ?
(1) 172 m
(2) 173 m
(3) 174 m
(4) 175 m
3/2
)
3 + 1)
2
3 −1
2
)
3 +1
2
6+ 2
)
AB is a vertical pole. The end A is on the level
ground. C is the middle point of AB. P is a point
on the level ground. The portion BC subtends
an angle β at P. If AP = n AB, then tan β =
(1)
(3)
n
2n + 1
2
n
n +1
2
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(2)
n
n −1
2
(4) none of these
Ashwani Goyal’s Tutorial
Mathematics - Material
12.
a man in a boat rowing away from a cliff 150
metres high observes that it takes 2 minutes to
change the angle of elevation of the top of the
cliff from 60° to 45°. The speed of the boat is
(
)
(2) (1/ 2 ) (9 + 3 3 ) km / h
(3) (1/ 2 ) (9 3 ) km / h
(1) (1/ 2 ) 9 − 3 3 km / h
14.
(1) ( a / 2 ) cos ec A tan α
16.
(3)
2 ab
a+b
(4)
2 ab
a−b
The upper three-quarters of a vertical pole
subtends an angle tan–1 (3/5) at a point in the
horizontal plane through its foot and distant 40
m from it. The height of the pole is
(1) 80 m
(2) 100 m
(3) 160 m
(4) 200 m
PQ is avertical tower and A, B, C are three
points on a horizontal line through Q, the foot of
the tower and on the same side of the tower. If
the angles of elevation of the top of the tower
from A, B and C are α , β , γ respectively, then
AB / BC =
(1)
(2)
(3)
(4)
ABCD is a rectangular park with AB = a. A
tower standing at C makes angle α and β at A
and B respectively, the height of the tower is
From the top of a building of height h, a tower
standing on the ground is observed to make an
angle θ . If the horizontal distance between the
building and the tower is h, the height of the
tower is
(1)
2h cos θ
sin θ + cos θ
(2)
2h
1 + cot θ
(3)
2h
1 + tan θ
(4)
2h
sin θ + cos θ
(1)
20.
sin θ =
: 16 :
(2)
2
cot α − cot 2 β
2
a cot α cot β
(4)
cot 2 α − cot 2 β
Two circular path of radii a and b intersect at a
point O and aB is a common chord of these
circles at A and B respectively. Chords OA and
OB subtend equal angles of 60° at their
repective centres. A vertical pole at O subtends
angle α and β respectively at A and B then
height of the pole is
(3)
21.
2
tan β − tan α
2
(1) − cot α
The angles of elevation of the top of a tower
standing on a horizontal plane, from two points
on a line passing through its foot at distance a
and b, respectively, are complementary angles.
If the line joining the two points subtends an
angle θ at the top of the tower, then if a > b
a
cot α + cot β
2
a tan α tan β
(3)
(4) ( a / 2 ) s ec α tan A
15.
a+b
a −b
a
Go
ya
(3) ( a / 2 ) cosec α cot A
19.
l's
(2) ( a / 2 ) sec A tan α
18.
(2)
h
(4) none of these
A person standing on the bank of a river
observes that the angle subtended by a tree on
the opposite bank is 60°, when he retires 40
metres from the banks he finds the angle at 30°.
The breadth of the river is
(1) 40 m
(2) 60 m
(3) 20 m
(4) 30 m
The elevatio of the top of a mountain at each of
the three angular points A, B and C of a plane
horizontal triangle is α , if BC = a the height of
the mountain is
a−b
a+b
M
at
13.
17.
(1)
a+b
cot α + cot β
(2) b cot β
(4) none of these
Three poles of height a, b, c stand on the same
side of road and subtend an angle of 45° at a
point on the line joining their feet. The pole of
height a subtends an angle α at the foot of the
pole of height b which subtends an angel β at
the foot of the pole with height c if a > b > c,
Copyright © 2013 GoyalsMath.com .All rights reserved.
Ashwani Goyal’s Tutorial
Mathematics - Material
then cot α − cot β =
(3)
A tower stands at the foot of a hill whose
inclination to the horizon is 9°; at a point 40 m up
the hill the tower subtends at angle of 54°. The
height of the tower is
(1) 17.56 m
(2) 45.76 m
(3) 54.76 m
(4) none of these
An aeroplane flying horizontally 1 km above the
ground is observed at an elevation of 60°. If
after 10 seconds, the elevation is observed to be
30°, then the uniform speed of the aeroplane
per hour is
(1) 120 km
(2) 240 km
(3) 240 3 km
24.
(4) 240 / 3 km
The angle of elevation of a stationary clound
from a point 2500 metres above a lake is 15 °
and the angle of depression of its reflection in
the lake is 45°. The height of the cloud above
the lake level is
28.
(3) 2500 3 m
29.
30.
(2) 2500 m
(4) 5000 3 m
(4) tan 2 3
sin α + cos β
sin α
(4)
sin α + sin β
cos α
Three poles whose feet lie on a circle subtend
(1) A.P.
(2) G.P.
(3) H.P.
(4) none of these
A person stands at a point A due south of a
tower and observes that its elevation is 60°. He
then walks westwards towards B, where the
elevation is 45°. At a pint C on AB produced, he
finds it to be 30°. then AB/BC is equal to
(1) 1/2
(2) 1
(3) 2
(4) 5/2
a pole standa at pont A on the boundary of a
circular park of radius a and subtends an angle
α at another point B on the boundary. If the
chord AB subtends an angle α at the centres
of the path, the height of the pole is
(4) 2a cos (α / 2 ) cot α
31.
A, B, C are three points on a vertical pole
whose distances from the foot of the pole are in
A.P. and whose of elevation at a point on the
ground are α , β and γ
at A, B,C respectively, then
respectively. If
α + β + γ = π , then tan α and γ is equal to
+ c 2 (cot α − cot β ) is equal to
(2) 0
(4) a + b + c
(3)
(3) 2a sin (α / 2 ) tan α
a 2 ( cot β − cot γ ) + b 2 ( cot γ − cot α )
(1) – 1
(3) 1
sin α + sin β
cos β
(2) 2a sin (α / 2 ) cot α
−1
A tower PQ stands at a pint P with in the
triangular park ABC such that the sides a, b,c
of the triangle subtends equal anglea at P, the
foot of the tower and the tower subtends anglea
α , β ,γ
(2)
(1) 2a cos (α / 2 ) tan α
a tower throws a shoadow of 2 3 metres
along the ground then the angle (in degrees)
that the sun makes with the ground is
(1) 15°
(2) 0°
26.
sin α + cos β
sin β
cot α , cot β , cot γ in
If a flagstaff 6 metres high placed on the top of
(3) 60°
(1)
angle α , β , γ respectively at the centre of the
circle. If the height of the poles are in A.P. then
Go
ya
(1) 2500 / 3 m
25.
ac − b
bc
balloon at A then tan θ is equal to
2
M
at
23.
(4)
If θ is the elevation of the highest point of the
l's
22.
ab − c
bc
man’s eye and the elevation of its centres is β .
bc − a 2
(2)
ab
2
A spherical balloon subtends an angle 2α at
h
ac − b 2
(1)
ab
27.
32.
(1) 3
(2) 2
(3) 1
(4) –1
A, B, C are three pionts on a horizontal line
through the base O of a pillar OP, such that OA,
OB, OC in A.P. If α , β , γ
: 17 :
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the angles of
Ashwani Goyal’s Tutorial
Mathematics - Material
(
2
2
2
2
(4) cot α − cot δ = 3 cot α − cot γ
elevation of the top of the pillar at A, B,C
respectively are also in a.P. then
37.
sin α ,sin β , sin γ are in
(1) 1
(3)
34.
(2) 1/ 3
(4)
3
then cos (α + β ) =
(1) 5 / 26
3/2
The angle of elevation of the top Q of a tower
PW at a point a on the horizontal plane through
P the foot of the tower is α . At a point of B on
AQ at a vertical height of a, the angle of
elevation of the middle point R of the tower PQ
an angle β at Q, the middle point of BC. If PQ
subtends
M
at
39.
l's
Go
ya
(2) sec α tan β
(3) tan α sec β
(4) none of these
)
40.
(
)
β − cot δ = 3 (cot α − cot γ )
γ − cot δ = 3 (cot α − cot β )
2
2
2
(2)
)
(a
2
− b 2 ) 2h 2
(
)
2
2
2
(4) 2 a + b h
A vertical tower OP of height h subtends angle
α , β , γ respectively at the points A, B, C on
the horizontal plane through the foot O of the
tower. A is due west of the tower. B is due east
of A and on the same side of the tower as A. A
is due south of B, AC =
PQ and RS are two vertical towers of the same
height. The line joining the top P and the foot S
of the two towers meets the horizontal line
through Q at a point A where the angles of
elevation of the tops P and R of the two towers
is α and β repectively. If AS = a, the height of
the towers is
2
2
2
2
(1) cot β − cot γ = 3 cot α − cot δ
2
then
(4) h cot 2 γ + cot 2 β − 2 cot α cot β
of time are α , β , γ and δ
2
A,
(3) h cot 2 α + cot 2γ − 2 cot α cot β
From a pint on the ground, if the angle of
elevation of a bird flying at constant speed in
horizontal direction, measured at equal intervals
2
at
(2) h cot 2 γ − cot 2 β
the boundary where AB subtends an angle 2 β
at the foot of the lamppost. If γ is the angle
which the lamppost subtends at C, the middle
point of the line joining a and B, then tan γ =
(1) tan α tan β
θ
(1) h ( cot α − cot β )
A lamppost stands in the centre of a circular
garden and makes angle α at pint A and B on
2
(3) cot
angle
cot −1 β cos 2 θ = k 2 , where k=
(
2a ( tan α cot β − 1)
2 tan α cot β − 1
2
(2) cot
an
2
2
2
(3) 2h / a = b
2a ( tan α tan β − 1)
(3)
2 tan α cot β − 1
36.
(3) 23 / 650
(4) 1/ 26
ABCD is a rectangular field with AB = a and
BC = b. A lamp oost of height h at A subtends
an angle α at P, the middle piont of CD and
another lamp post of equal height at D subtends
(
2a ( tan α − 2 tan β )
(2)
tan α − tan β
(4)
(2) 24 / 650
2
2
2
(1) a + b 2h
2a ( tan α − tan β )
(1)
tan α − 2 tan β
35.
α and β respectively at A where OA = 2m
38.
is β , then the height of the tower is
A vertical tower standing at O has marks
P,Q,R,S at heights of 1 m 2 m, 3 m and 4 m from
the foot O and A is a point on the horizontal
plane through O. If PQ and RS subtend angles
h
33.
(1) A.P.
(2) G.P.
(3) H.P.
(4) none of these
A ladder rests against a wall at an angle of 35°.
Its is pulled away through a distance a, so that it
slides a distance b down the wall, finally
makening an angle 25° with the horizontal, then
a/b =
)
(1)
(3)
: 18 :
a sin ( β − α )
cos β
a sin ( β − α )
sin β
Copyright © 2013 GoyalsMath.com .All rights reserved.
(2)
(4)
a cos ( β − α )
cos β
a cos ( β − α )
sin β